Find the mean and the median for the following frequency distribution:
| $x$ | $8$ | $9$ | $10$ | $11$ | $12$ |
|---|---|---|---|---|---|
| $f$ | $5$ | $4$ | $2$ | $6$ | $3$ |
| $x_i$ | $f_i$ | $f_i x_i$ | Cumulative frequency |
|---|---|---|---|
| $8$ | $5$ | $40$ | $5$ |
| $9$ | $4$ | $36$ | $9$ |
| $10$ | $2$ | $20$ | $11$ |
| $11$ | $6$ | $66$ | $17$ |
| $12$ | $3$ | $36$ | $20$ |
$\Sigma f_i = N = 20$ (Even), $\;$ $\Sigma f_i x_i = 198$
Mean $= \dfrac{\Sigma f_i x_i}{\Sigma f_i} = \dfrac{198}{20} = 9.9$
$\because$ $N$ is even, median $= \dfrac{\left(\dfrac{N}{2}\right)^{th} \text{term} + \left(\dfrac{N}{2} + 1\right)^{th} \text{term}}{2}$
i.e. $\;$ Median $= \dfrac{\left(\dfrac{20}{2}\right)^{th} \text{term} + \left(\dfrac{20}{2} + 1\right)^{th} \text{term}}{2}$
i.e. $\;$ Median $= \dfrac{10^{th} \; \text{term} + 11^{th} \; \text{term}}{2}$
i.e. $\;$ Median $= \dfrac{10 + 10}{2} = \dfrac{20}{2} = 10$